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Simplifying 3b2 + -12b = 23 Reorder the terms: -12b + 3b2 = 23 Solving -12b + 3b2 = 23 Solving for variable 'b'. Reorder the terms: -23 + -12b + 3b2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + -12b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -7.666666667 + -4b + b2 = 0 Move the constant term to the right: Add '7.666666667' to each side of the equation. -7.666666667 + -4b + 7.666666667 + b2 = 0 + 7.666666667 Reorder the terms: -7.666666667 + 7.666666667 + -4b + b2 = 0 + 7.666666667 Combine like terms: -7.666666667 + 7.666666667 = 0.000000000 0.000000000 + -4b + b2 = 0 + 7.666666667 -4b + b2 = 0 + 7.666666667 Combine like terms: 0 + 7.666666667 = 7.666666667 -4b + b2 = 7.666666667 The b term is -4b. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4b + 4 + b2 = 7.666666667 + 4 Reorder the terms: 4 + -4b + b2 = 7.666666667 + 4 Combine like terms: 7.666666667 + 4 = 11.666666667 4 + -4b + b2 = 11.666666667 Factor a perfect square on the left side: (b + -2)(b + -2) = 11.666666667 Calculate the square root of the right side: 3.415650255 Break this problem into two subproblems by setting (b + -2) equal to 3.415650255 and -3.415650255.Subproblem 1
b + -2 = 3.415650255 Simplifying b + -2 = 3.415650255 Reorder the terms: -2 + b = 3.415650255 Solving -2 + b = 3.415650255 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + b = 3.415650255 + 2 Combine like terms: -2 + 2 = 0 0 + b = 3.415650255 + 2 b = 3.415650255 + 2 Combine like terms: 3.415650255 + 2 = 5.415650255 b = 5.415650255 Simplifying b = 5.415650255Subproblem 2
b + -2 = -3.415650255 Simplifying b + -2 = -3.415650255 Reorder the terms: -2 + b = -3.415650255 Solving -2 + b = -3.415650255 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + b = -3.415650255 + 2 Combine like terms: -2 + 2 = 0 0 + b = -3.415650255 + 2 b = -3.415650255 + 2 Combine like terms: -3.415650255 + 2 = -1.415650255 b = -1.415650255 Simplifying b = -1.415650255Solution
The solution to the problem is based on the solutions from the subproblems. b = {5.415650255, -1.415650255}
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